Origin of the Mode-Splitting in a Microwave Sapphire Whispering-Gallery Mode Resonator

Cylindrical whispering-gallery mode (WGM) resonators, machined from high-quality sapphire mono-crystal and cooled to liquid helium temperature, exhibit exceptionally high <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula>-factors in the microwave frequency domain. Such resonators serve as the core for ultra-stable oscillator featuring fractional frequency stability exceeding <inline-formula> <tex-math notation="LaTeX">$1 \times 10^{-15}$ </tex-math></inline-formula> at short integration times. Similar to any cylindrical resonant structure, the WGM resonator exhibits a twofold degeneracy. When a defect disrupts the cylindrical symmetry of the resonator, the WGMs split and appear as doublets. In the high-quality sapphire resonators, the frequency separation between these twin modes varies from one mode order to another, reaching a maximum value of a few tens of kilohertz. While mode splitting for a given mode was previously deemed unpredictable and intrinsic to each resonator, assumed to result from randomly distributed defects, our findings indicate that the observed mode splitting in all sapphire resonators, regardless of their origin, primarily arises from a common defect due to the different response of the sapphire’s crystallographic plans to the manufacturing processes.